Low-frequency lattice vibrations from atomic displacement parameters of α-FOX-7, a high energy density material

The normal mode analysis of variable-temperature anisotropic atomic displacement parameters (ADPs) of the α-phase of 1,1-diamino-2,2-dinitroethylene (DADNE or FOX-7) is reported.


Introduction
FOX-7 (DADNE, C 2 H 4 N 4 O 4 ) (Latypov et al., 1998;Trzciń ski & Belaada, 2016) is a high energy density material (HEDM) that shows a set of phase transformations as a function of temperature and/or pressure (Bu et al., 2020). The structural and mechanical stability of different polymorphs are defined, among other factors, by their thermal molecular vibrations. The structures of all reported crystalline forms of -,and -FOX-7 are built from wave-shaped layers of FOX-7 molecules linked by hydrogen bonding and the stacked layers are weakly bound by van der Waals interactions (Crawford et al., 2007).
The mechanisms for the impact-induced initiation of HEDMs has been debated for many decades. Understanding the initiation process is essential for designing safe and wellperforming materials suitable for use across military and civilian applications. Building on early numerical models (Tokmakoff et al., 1993), recent theoretical developments have suggested that initiation is strongly correlated to the dynamic behaviour of the material (Michalchuk et al., 2019(Michalchuk et al., , 2021a. When struck by a mechanical force, the energy is inserted into the lattice vibrations, up-converting through phonon-phonon collisions until the molecules are vibrationally excited. If sufficiently excited, covalent-bond rupture occurs, leading to the primary initiation event. These so-called 'phonon uppumping' models have proved to be very promising across a wide range of HEDMs, though their further development requires detailed investigations into the structural dynamics of these materials. In connection with this, FOX-7 is of particular interest, owing to its polymorphic behaviour and layered crystal packing, the latter widely believed to indicate insensitivity to impact initiation (Ma et al., 2014). Recent studies have suggested that FOX-7 may undergo a polymorphic transformation in response to mechanical impact (Michalchuk et al., 2021b), although the influence of such transformations on material performance is not known. Within the framework of phonon up-pumping, a thorough understanding of FOX-7 dynamics is needed before its complex initiation behaviour can be further elucidated.
The use of neutron or X-ray inelastic scattering for studying vibrational properties is rather limited for FOX-7 due to the low molecular symmetry and high lattice anharmonicity, and hence relatively large crystals of FOX-7 are required. This is why only phonon density of states measured with neutron inelastic scattering from powders have been reported so far (Hunter et al., 2015;Michalchuk et al., 2019). Moreover, the presence of H atoms represents a problem for coherent neutron scattering experiments. Raman and IR spectroscopies provide information on the vibrational frequencies at the À-point of the Brillouin zone and are typically restricted to the analysis of wavenumbers > 50-100 cm À1 ; data on low-frequency vibrations and dispersion are very limited. The majority of spectroscopic techniques are focused on vibrational frequencies (eigenvalues of the dynamical matrices), while phonon eigenvectors are characterized experimentally for very few structurally simple materials (Strauch & Dorner, 1986;Pawley et al., 1980). Here we have applied a concurrent analysis of the variable-temperature ADPs routinely derived from diffraction experiment (Bü rgi & Capelli, 2000) to investigate the lattice vibration properties of the monoclinic -phase of FOX-7. This approach considers eigenvectors of dynamical matrices, which are encoded in the shape of displacement ellipsoids -smearing of atoms in the diffraction experiment. The phonon modes are modelled with line spectra as Einstein oscillators. For a molecular material, thermal vibrations are split on the rigid unit modes (RUMs)displacements of a molecule as a whole (translations and librations), and deformations associated with optic phonons that normally have higher frequencies. While this approximation is appropriate for all molecular solids, particularly where 'soft' molecular modes exist, such as -NO 2 wags, it holds well for crystalline FOX-7 (Michalchuk et al., 2021a,b).
The monoclinic -phase of FOX-7 is stable below 380 K. Thus far, only one set of single-crystal X-ray (Mo K) diffraction data containing five temperature points in the range 200-293 K at ambient pressure, to d min of 0.76 Å , is available (Evers et al., 2006). Additional powder data at 403 and 423 K have also been collected, indicating the first-order ! phase transition (monoclinic P2 1 /n ! orthorhombic P2 1 2 1 2 1 ) at 389 K (Evers et al., 2006). The out-of-plane displacements of four O atoms in two -NO 2 groups are clearly observed in both phases of FOX-7 in the range 200-423 K (Evers et al., 2006). More attention has been paid to FOX-7 under high pressure-temperature conditions, combining experiment and simulation to study its phase transitions, structural changes and vibrational behaviour (Peiris et al., 2004;Hu et al., 2006;Bishop et al., 2012;Dreger et al., 2013Dreger et al., , 2014Appalakondaiah et al., 2014;Hunter et al., 2015). Here we complement the available information on thermal vibrations with low-energy frequencies (librations and translations) and their anharmonic behaviour parameterized with Grü neisen parameters. We also include in the analysis some of the vibrations associated with deformation of the FOX-7 molecule, providing results that are reasonably close to the reported values for low-frequency optic phonons at the À-point of the Brillouin zone, as expected for the phonon modes with low dispersion.
The collection of diffraction data with synchrotron light can be easily done with very small crystals; high intensity and fast detectors reduce the data collection time to tens of seconds and the data can be collected with very fine temperature sampling. However, the data quality frequently suffers from a nonhomogeneous and/or unstable beam, an irregular shape of the crystal or inadequately characterized attenuation of the incoming and scattered beams with crystal mounts; all these effects are believed to be minimized with empirical absorption and scaling corrections. ADPs are the most sensitive parameters and may therefore contain an additional contribution not related to thermal smearing but rather linked to the data and data processing, as demonstrated by the simultaneous analysis of multi-temperature ADPs of the three glycine polymorphs (Aree & Bü rgi, 2012;Aree et al., 2013Aree et al., , 2014. Here we show that those contributions, being temperature independent, do not distort information on the low-energy thermal vibrations. To improve information concerning the dynamics of H atoms and general quality of ADPs, we have applied a nonspherical refinement of X-ray diffraction data developed by Kleemiss et al. (2021). Altogether, the data collection, data processing and structure refinement applied here show that a vibrational analysis similar to that presented here might become a relatively simple-to-use tool that offers unique information and can be easily implemented for singlecrystal diffraction experiments at synchrotron beamlines.

2.2.
Multi-temperature single-crystal X-ray diffraction 2.2.1. Synchrotron diffraction experiment. A single crystal of -FOX-7 (0.2 Â 0.1 Â 0.1 mm) was selected and mounted on the tip of a glass fibre with the minimum of high-temperature epoxy. Variable-temperature single-crystal diffraction data were collected at the Swiss-Norwegian Beamline BM01 (ESRF, Grenoble) (Dyadkin et al., 2016) upon heating from 80 to 360 K, with temperature control by an Oxford Cryosystems Cryostream cooler (Cosier & Glazer, 1986). Fifty eight temperature data points with a 4-8 K increment were  Table 1 Refinement statistics for -FOX-7 from spherical (XL) and nonspherical (NoSpherA2) refinements .   collected. For each temperature, a full data collection was carried out at a wavelength of 0.62379 Å (19.876 keV) with a single !-scan. Moreover, multi-temperature data were continuously collected to 474 K and the data between 274 and 474 K were used for an investigation of the thermal expansion and phase transitions of the energetic material FOX-7 (McMonagle et al., 2022). This work shared data between 80 and 360 K for the normal mode analysis. 2.2.2. Data processing. The data were processed with CrysAlis PRO (Rigaku OD, 2016) and the structures were refined with SHELXL (Sheldrick, 2015) in a sequential manner as described in Chernyshov et al. (2019) and Bogdanov et al. (2021). After inspecting the temperaturedependent ADPs of -FOX-7 and removing the outliers, we decided to use 10 temperature points with 20-40 K steps, which adequately define the continuous smooth ADP curves in the range 80-360 K for the normal mode analysis. OLEX2 (Dolomanov et al., 2009;Bourhis et al., 2015) was then used for the refinement with nonspherical scattering functions (Kleemiss et al., 2021). Note that in the range 80-360 K, the diffraction full and max values of 23.49 and 32.72-32.97 yield respective data coverages of 89.3-90.9 and 66.0-67.3%. Although the data extended to sin / ' 0.8, the I/ ratios in the outer shells were rather poor, in particular at T > 280 K (Table 1).
2.2.3. Simulation of C-point a-FOX-7 unit cell. Periodic plane-wave density functional theory (DFT) simulations were performed in CASTEP (Version 20.11; Clark et al., 2005). The electronic structure was expanded in plane waves to a kinetic energy cut-off of 1200 eV, with a charge-density cut-off of 35.49 Å À1 . The exchange correlation functional of Perdew-Burke-Ernzerhof (PBE) (Perdew et al., 1996) was used, alongside the semi-empirical dispersion correction of Tkatchenko-Scheffler (TS) (Tkatchenko & Scheffler, 2009). The electronic wavefunction was accepted following convergence < 10 À13 eV and the residual forces converged < 10 À4 eV per atom. À-point frequencies and eigenvectors were simulated through the linear response method (Refson et al., 2006), without explicit consideration for LO-TO splitting. Dynamical matrices were subsequently calculated on a 3 Â 3 Â 3 Monkhorst-Pack grid (Monkhorst & Pack, 1976), and interpolated onto a 9 Â 9 Â 9 fine grid, with which the ADPs were calculated as implemented within the CASTEP suite.

DFT calculations of internal vibrational frequencies.
The atomic coordinates of -FOX-7 at 80 K were employed to estimate internal vibrational frequencies. The structure was initially optimized with the semi-empirical PM3 method and was then fully re-optimized using DFT calculations in the gas phase at the B3LYP/6-311+G(2d,p) level of theory with the program GAUSSIAN09 (Frisch et al., 2009). The energy minimization converged smoothly to a global minimum. After scaling, the harmonic vibrational frequencies in the range 57-3546 cm À1 agree with the literature data (see x3.3).

Results and discussion
3.1. Crystal structure of a-FOX-7 There are four -FOX-7 molecules in the monoclinic unit cell with the space group P2 1 /n [ Fig. 1(b)]. The -FOX-7 molecule is nonplanar, as indicated by the larger deviations (Å ) of atoms from the mean molecular plane: O11 À0.209 (1), O21 À0.441 (1) and O22 0.761 (1), and the greater variations ($9-37 ) of the C2-C1-N11(N12)-O11(O21/O22) torsion angles from planarity for the 80 K data [ Figs. 1(a) and 1(b)]. This is due to the steric hindrance between the two nitro groups and the small number of hydrogen-bonding interactions [ Fig. 1(d)]. In the crystal, adjacent -FOX-7 molecules are closely connected via N-HÁ Á ÁO hydrogen bonds along the c axis, forming herringbone layers with an obtuse interplanar angle of 140.10 (2) [ Fig. 1(c)]. These layers are loosely packed along the b axis, allowing greater changes on this axis, as observed from the unit-cell volume expansion with increasing temperature (Fig. 2) and from the unit-cell volume contraction at high pressure (Hunter et al., 2015).

Multi-temperature ADPs of a-FOX-7
The multi-temperature ADPs of -FOX-7 behave as expected within the harmonic approximation; see the principal elements U 11 , U 22 and U 33 for atoms C1, N21 and O11 [ Figs. 3(a), 3(b) and 3(c), respectively]. In the classical regime, the ADPs increase linearly with temperature (80-164 K), but begin to increase more steeply at higher temperatures, thereby indicating marked lattice anharmonicity in -FOX-7. This anharmonicity is captured by the Grü neisen parameter (see x3.4). As the -FOX-7 data do not cover the quantum regime (low temperature-independent limit), the theoretically temperature-independent ADPs, observed in the glycine polymorphs (Aree et al., 2014), are not noticed here. Note that the elements U 11 of all atoms increase more slowly at T > 300 K, resulting in an intersection of the curves U 11 and U 33 . This is probably due to the approach of the -tophase transition at 389 K (Evers et al., 2006), although the unit-cell parameters of -FOX-7 do not show a discontinuity at the temperature of the present experiment (Fig. 2). The data resolution is not sufficient to really see the effects of bonding, so that the more elaborate charge-density description (NoSpherA2) (Table 1).
We attempted to reproduce the variable-temperature ADPs of -FOX-7 using periodic DFT simulations at the PBE-TS level of theory (Fig. 4). At the fully optimized geometry (i.e. the 0 K structure), our simulated harmonic ADPs consistently underestimate the magnitude of the primary displacement vectors, even at 100 K. This effect is, however, relatively small for both U 22 and U 33 . This indicates a significant degree of anharmonicity in the FOX-7 structure, particularly in the direction between herringbone chains. Within the harmonic model, the mean-square atomic displacement increases approximately linearly with temperature. Thus, the growing deviation between our simulation from experiment with Multi-temperature ADPs of -FOX-7 for atoms (a) C1, (b) N21 and (c) O11 from XL and NoSpherA2 refinements. The standard uncertainties are 3 Â 10 À4 Å 2 , or ca the line thickness. The displacement ellipsoid plot (50% probability level) with atom numbering is shown for -FOX-7 at 80 K.

Figure 4
The average absolute deviations of ADPs for each atom type in -FOX-7, with ÁU ¼ 1 for the i atoms of each type. Values are shown as absolute deviations between the ADPs from the harmonic simulation and the diffraction experiment at each temperature. temperature is expected and consistent with the increased anharmonicity as the temperature rises. There is an intriguing divergence of U 33 for all three atom types, occurring at ca 250 K. As observed in the experimental ADPs (Fig. 3), there is a rapid increase in motion along this direction at this temperature. As this direction corresponds to motion along the hydrogen-bonded herringbone chains, we can suggest that this increased divergence presumably reflects a weakening of the hydrogen-bonded chains with temperature. Further and dedicated efforts are ongoing to analyse this peculiar feature.

Internal vibrations from DFT calculations
Upon scaling by a factor of 0.965, the 36 harmonic vibrational frequencies of -FOX-7 obtained from the DFT/ B3LYP/6-311+G(2d,p) calculation in the vacuum range from 57 to 3546 cm À1 agree overall with those from the MP2/6-31G(d,p) method (Sorescu et al., 2001), the periodic DFT calculations using CASTEP (Averkiev et al., 2014;Su et al., 2019) and Raman spectroscopy   (Table 2). The three lowest frequencies (57, 92 and 115 cm À1 ) overlap with lattice frequencies and correspond to NO 2 torsion, skeleton deformation and NH 2 wagging modes, respectively. The 33 higher internal vibration frequencies (203-3546 cm À1 ) were included for the calculation of the anisotropic temperature-independent contributions " to the ADPs for H atoms, which were constrained in the normal mode analysis (Table 3). Note that the larger value of " 33 for H atoms is mainly attributed to the out-of-plane motions of higher frequencies (203-378 cm À1 ).

Crystal dynamics of a-FOX-7 from normal mode analysis
We have two sets of -FOX-7 ADPs deduced from spherical and nonspherical refinements with the respective programs SHELXL (XL) and OLEX2 -NoSpherA2 (NoSph). There are three models of motions for parameterizing the variabletemperature ADPs. (i) Model rbeg stands for a typical rigidbody motion with three translations (T x , T y and T z ) and three librations (L x , L y and L z ), a Grü neisen constant for each of the six frequencies and two epsilons (the temperature-independent ADPs), each for the H and non-H atoms. (ii) Model rbeg+3b explicitly indicates the addition of three bending Table 2 Comparison of internal vibrational frequencies (cm À1 ) of -FOX-7 from calculations and Raman measurement. deformations of NO 2 and CN 2 groups (U 1 , U 2 and U 3 ) to rbeg.
(iii) Model rbeg+3b+1f further includes one temperatureindependent high frequency, which is attributed to CN 2 wagging and NO 2 twisting. The molecular orientation is set as follows: the x axis passing through the N22!O11 vector is completed with a right-hand rule by the y axis going through the N21!O22 vector (Fig. 5). The results of normal mode analysis are summarized in Table 3. The lattice vibrational frequencies from ADP analysis are compared to those derived from other techniques in Table 4. The model of motion rbeg+3b+1f provides estimated ADPs in fair agreement with the ADPs from diffraction, as depicted with the quite random  Table 3 Normal mode analysis of multi-temperature ADPs of -FOX-7.
Clearly, the rbeg model is insufficient to describe the large out-of-plane motions of the NO 2 groups in -FOX-7, as indicated by the rather high values of GOF > 5% and wR 2 >> 10% for both sets of ADPs (Table 3). The model of motion is significantly improved by the addition of the deformations arising from bending, wagging and twisting of NO 2 and CN 2 groups, as evidenced from the values of GOF = 3.19 and wR 2 = 9.30%. The six lattice vibrational frequencies (translations: 39.4, 44.6 and 56.6 cm À1 ; librations: 76.5, 85.2 and 97.5 cm À1 ) and one deformation frequency (145.5 cm À1 ) obtained are in line with those derived theoretically and spectroscopically, i.e. DFT-D and INS (Hunter et al., 2015), and from Raman spectroscopy ) (see Table 4). Moreover, the Grü neisen parameters (2.1-2.5) from ADP analysis are similar to those deduced from periodic Hartree-Fock calculations, i.e. 2.5 at 75 K and 1.0 at 300 K (Zerilli & Kuklja, 2007) and from temperature-pressure-variable synchrotron diffraction experiments, i.e. 1.1 at ambient conditions (Zhang et al., 2016). Note that if the six Grü neisen parameters were refined independently, the translational and librational frequencies obtained are mostly intact. The exception is the change of $15 cm À1 , of which the libration 0.7L x -0.7L z is compensated by the translation 0.6T x -0.8T y + 0.1T z . The values of GOF = 2.50 and wR 2 = 7.23% are improved, but the six Grü neisen parameters vary greatly, i.e. in the range 0.7-8.8 (Table 3). This suggests anharmonicity of the in-layer and out-of-layer vibrations due to the highly anisotropic thermal expansivities of -FOX-7.

Conclusions
1,1-Diamino-2,2-dinitroethylene, also known as DADNE or FOX-7, is an insensitive highly explosive material. Under ambient conditions, FOX-7 exists in an -phase and it transforms to theand -forms at high-temperature-pressure due to the distinct 3D molecular arrangements, although the three phases are similarly constructed from herringbone layers of nonplanar molecules due to the large out-of-plane deviations of the nitro O atoms. The low-frequency lattice vibrations, together with a complete set of X-ray diffraction data of -FOX-7 covering a large temperature range (80-360 K), remain elusive. We therefore applied Bü rgi's method of concurrent analysis of multi-temperature atomic displacement parameters (ADPs) from diffraction data to explore the crystal dynamics of crystalline -FOX-7.
Due to the abundance of intermolecular N-HÁ Á ÁO hydrogen bonds in -FOX-7, the ADPs are minimally biased by the valence electron density. Hence, the ADPs derived from nonspherical refinement (NoSpherA2) with OLEX2 (Dolomanov et al., 2009;Bourhis et al., 2015) are slightly decreased when compared to those from conventional spherical refinement with SHELXL. The variable-temperature nonspherical ADPs are suitably parameterized by a model of motion rbeg+3b+1f, which includes a typical rigid-body motion, a Grü neisen constant, two epsilons (the temperatureindependent ADPs for H and non-H atoms), three bending deformations of NO 2 and CN 2 groups, and one temperatureindependent high frequency (attributed to CN 2 wagging and NO 2 twisting). The anharmonicity arising from in-layer and out-of-layer vibrations is parameterized by the distinct Grü neisen parameters. In addition, we demonstrate that despite the limited quality of the diffraction data, the lattice vibrational frequencies from ADP analysis are reasonably close to those derived from inelastic scattering, Raman measurements and DFT calculations.
We conclude with a general note on experimentation. Single-crystal data collection with bright synchrotron radiation is fast, but complete and highly redundant 'multi-run' high-resolution data may allow sufficient time to see the effects associated with beam instabilities and radiation damage. Single-run data acquisitions, like those used here, rapidly map the temperature evolution of a crystal structure but suffer from reduced completeness and redundancy. With the example of vibrational analysis based on temperaturedependent ADPs, we show that the useful and sometimes unique information content does not suffer from such a compromise.